* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
cond(true(),y) -> y
- Signature:
{admit/2,cond/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit,cond} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3()
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3()
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
cond(true(),y) -> y
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3()
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3()
- Strict TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(.) = {2},
uargs(cond) = {2},
uargs(cond#) = {2},
uargs(c_1) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(.) = [1] x1 + [1] x2 + [0]
p(=) = [0]
p(admit) = [4] x1 + [6] x2 + [1]
p(carry) = [2]
p(cond) = [1] x2 + [0]
p(nil) = [0]
p(sum) = [1] x1 + [1] x2 + [1] x3 + [0]
p(true) = [0]
p(w) = [2]
p(admit#) = [1] x1 + [6] x2 + [0]
p(cond#) = [1] x2 + [0]
p(c_1) = [1] x1 + [2]
p(c_2) = [0]
p(c_3) = [0]
Following rules are strictly oriented:
admit(x,.(u,.(v,.(w(),z)))) = [6] u + [6] v + [4] x + [6] z + [13]
> [1] u + [1] v + [6] z + [11]
= cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) = [4] x + [1]
> [0]
= nil()
Following rules are (at-least) weakly oriented:
admit#(x,.(u,.(v,.(w(),z)))) = [6] u + [6] v + [1] x + [6] z + [12]
>= [1] u + [1] v + [6] z + [13]
= c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) = [1] x + [0]
>= [0]
= c_2()
cond#(true(),y) = [1] y + [0]
>= [0]
= c_3()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: PredecessorEstimation WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3()
- Weak TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,3}
by application of
Pre({1,2,3}) = {}.
Here rules are labelled as follows:
1: admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
2: admit#(x,nil()) -> c_2()
3: cond#(true(),y) -> c_3()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
admit#(x,nil()) -> c_2()
cond#(true(),y) -> c_3()
- Weak TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
2:W:admit#(x,nil()) -> c_2()
3:W:cond#(true(),y) -> c_3()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: cond#(true(),y) -> c_3()
2: admit#(x,nil()) -> c_2()
1: admit#(x,.(u,.(v,.(w(),z)))) -> c_1(cond#(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z))))))
* Step 6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
admit(x,.(u,.(v,.(w(),z)))) -> cond(=(sum(x,u,v),w()),.(u,.(v,.(w(),admit(carry(x,u,v),z)))))
admit(x,nil()) -> nil()
- Signature:
{admit/2,cond/2,admit#/2,cond#/2} / {./2,=/2,carry/3,nil/0,sum/3,true/0,w/0,c_1/1,c_2/0,c_3/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {admit#,cond#} and constructors {.,=,carry,nil,sum,true,w}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))